Form and logic

In the beginning is an undifferentiated state, an unmarked space. The first distinction is the first differentiation, the advent of a mark, a cross, a form. The unmarked state is the urgrund of the form, its origin and basis. The marked state is the form of the mark, a cross into markedness. The form encapsulates the progression from undifferentiated unity to differentiated hierarchy.

Spencer-Brown applies the calculus of indications to propositional logic in his book Laws of Form (Julian Press, New York, 1972), Appendix 2. This requires an interpretation of the mark as either true or false. He acknowledges that the choice is arbitrary, and then takes the mark to represent truth. This corresponds to the common logical convention that a proposition is assumed to be false unless it is shown to be completely true.

True propositions can be conceived as arising islands of truth in a sea of falsehood; or they can be conceived as the sea itself, interrupted by islands of falsehood. They are opposite conventions, and there is no logical reason to prefer one or the other.

But they encode different conceptions of truth. Is truth rare or common? Is falsehood rare or common? Do we give a proposition the benefit of the doubt or accept it if it has some truth? Or do we reject all doubtful propositions or partially true propositions?

The form is pre-logical; it assumes no convention about truth values. The form also assumes no convention about classes. A logic of classes can adopt a convention that the mark is the universal class or the null class. It depends on whether classes are conceived as arising from a unmarked space of nullity or void, or from an unmarked space of everything or pleroma.

Laws of form easily fits universal forms of the logic of classes but existential sentences are problematic. Spencer-Brown observes this interpretative theorem:

An existential inference is valid only in as far as its algebraic structure can be seen as a universal inference.

This theorem holds for both conventions about classes, that is, whether existence is considered to stand out from a void or against a pleroma.

These dual conceptions lead to a distinction between sets, which are composed of members that may have certain attributes, and classes, which are composed of attributes that may have certain members. The null set is a set with no members. The universal set has all members within a given context. The universal class has no attributes. The null class has all attributes within a given context.